Theorem nfbidf 1774

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)

Hypotheses

Ref	Expression
nfbidf.1	|- F/xph
nfbidf.2	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
nfbidf	|- (ph -> ( F/xps <-> F/xch))

Proof of Theorem nfbidf

Step	Hyp	Ref	Expression
1		nfbidf.1	. . 3 |- F/xph
2		nfbidf.2	. . . 4 |- (ph -> (ps <-> ch))
3	1, 2	albid 1772	. . . 4 |- (ph -> (A.xps <-> A.xch))
4	2, 3	imbi12d 311	. . 3 |- (ph -> ((ps -> A.xps) <-> (ch -> A.xch)))
5	1, 4	albid 1772	. 2 |- (ph -> (A.x(ps -> A.xps) <-> A.x(ch -> A.xch)))
6		df-nf 1545	. 2 |- ( F/xps <-> A.x(ps -> A.xps))
7		df-nf 1545	. 2 |- ( F/xch <-> A.x(ch -> A.xch))
8	5, 6, 7	3bitr4g 279	1 |- (ph -> ( F/xps <-> F/xch))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfsb4t  2080  dvelimdf  2082  nfcjust  2478  nfceqdf  2489